Download Path Loss Exponent Estimation for Wireless Sensor Network

1
Path Loss Exponent Estimation for Wireless Sensor
Network Localization
Guoqiang Mao, Brian D.O. Anderson and Barış Fidan
Abstract
The wireless received signal strength (RSS) based localization techniques have attracted significant research interest for their
simplicity. The RSS based localization techniques can be divided into two categories: the distance estimation based and the
RSS profiling based techniques. The path loss exponent (PLE) is a key parameter in the distance estimation based localization
algorithms, where distance is estimated from the RSS. The PLE measures the rate at which the RSS decreases with distance, and
its value depends on the specific propagation environment. Existing techniques on PLE estimation rely on both RSS measurements
and distance measurements in the same environment to calibrate the PLE. However distance measurements can be difficult and
expensive to obtain in some environments. In this paper we propose several techniques for online calibration of the PLE in
wireless sensor networks without relying on distance measurements. We demonstrate that it is possible to estimate the PLE using
only power measurements and the geometric constraints associated with planarity in a wireless sensor network. This may have a
significant impact on distance-based wireless sensor network localization.
I. I NTRODUCTION
Most wireless sensor network (WSN) applications require knowing or measuring locations of thousands of sensors accurately.
In environmental sensing applications such as bush fire surveillance, water quality monitoring and precision agriculture, for
example, sensing data without knowing the sensor location is meaningless [1]. In addition, location estimation may enable
applications such as inventory management, intrusion detection, road traffic monitoring, health monitoring, etc.
Sensor network localization estimates the locations of sensors with unknown location information with regards to a few
sensors with known location information by using inter-sensor measurements such as distance and bearing measurements. Intersensor measurement techniques in WSN localization can be broadly classified into three categories: received signal strength
(RSS) measurements, angle of arrival (AOA) measurements, and propagation time based measurements. Among them, RSS
based localization techniques have received considerable research interest [1]–[8]. Although RSS based localization techniques
can only provide a coarse-grained location estimate, they are attractive because of their simplicity. Received signal strength
indicator (RSSI) has become a standard feature in most wireless devices, therefore RSS based localization techniques require
no additional hardware, and are unlikely to significantly impact local power consumption, sensor size and thus cost. RSS based
localization techniques can be further divided into distance-estimation based [1], [3]–[6] and RSS-profiling based techniques
[2], [7], [9], [10].
Existing distance-estimation based techniques rely on a log-normal radio propagation model [11] to estimate inter-sensor
distances from RSS measurements. The path loss exponent (PLE) is a key parameter in the log-normal model. An accurate
knowledge of the PLE is required in order to obtain an accurate estimate of the inter-sensor distance from the corresponding
RSS measurement. Existing techniques either consider the PLE is known a priori by assuming the WSN environment is free
space, or obtain the PLE through extensive channel measurement and modeling by measuring both RSS and distances in
the same environment of WSN prior to system deployment. However the PLE is environment dependent. Even in the same
environment, the channel characteristics may change considerably over a long period of time due to seasonal changes and
weather changes [11]. Therefore it is an over-simplification to assume a free space environment. On the other hand a priori
channel measurement and modeling may not be possible for monitoring and surveillance applications in hostile or inaccessible
environments. In this paper we propose several techniques for online calibration of the path loss exponent, which do not rely
on distance measurements.
The RSS has been popularly modeled by a log-normal model [1], [11]–[13]:
2
Pij [dBm] ∼ N(Pij [dBm], σdB
),
(1)
Pij [dBm] = P0 [dBm] − 10 × α × log10 (dij /d0 ),
(2)
where Pij [dBm] is the received power at a receiving node j from a transmitting node i in dB milliwatts, Pij [dBm] is the
2
is the variance of the shadowing, P0 [dBm] is the received power in dB milliwatts at a
mean power in dB milliwatts, σdB
reference distance d0 , and dij is the distance between nodes i and j. In this paper, we use the notation [dBm] to denote
that power is measured in dB milliwatts. Otherwise, it is measured in milliwatts. The reference power P0 [dBm] is calculated
G. Mao is with the University of Sydney and National ICT Australia, Sydney. B. D. O. Anderson and B. Fidan are with Australian National University and
National ICT Australia, Canberra. National ICT Australia is funded by the Australian Governments Department of Communications, Information Technology
and the Arts and the Australian Research Council through the Backing Australias Ability initiative and the ICT Centre of Excellence Program.
2
using the free space Friis equation or obtained through field measurements at the reference distance d0 [11]. It is important
to note that the reference distance d0 should be in the far field of the transmitter antenna so that the near-field effect does
not alter the reference power. In large coverage cellular systems, a 1 km reference distance is commonly used, whereas in
microcellular systems, a much smaller distance such as 100 m or 1 m is used [11]. In this paper, it is assumed that d0 and
P0 [dBm] can be obtained from a priori calibration of the wireless device and they are known constants. In many situations,
Pij [dBm] 6= Pji [dBm] due to the asymmetric wireless signal propagation path from node i to node j and from node j to
node i. However in this paper, it is assumed that Pij [dBm] = Pji [dBm] for simplicity. In the case of an asymmetric path, the
average value of Pij [dBm] and Pji [dBm] is used [1]. It is also assumed that all distances are normalized with respect to d0 .
The parameter α is called the path loss exponent, which measures the rate at which the received signal strength decreases
with distance. The value of α depends on the specific propagation environment. In this paper, it is assumed that α is an unknown
constant, which needs to be determined. For a large wireless sensor network spanning a wide area with different environmental
conditions, the area can be divided into smaller regions and α can be considered as a constant in each region. Path loss exponent
estimation plays an important role in distance-based wireless sensor network localization, where distance is estimated from the
received signal strength measurements [14]. Path loss exponent estimation is also useful for other purposes like sensor network
dimensioning. As the value of the path loss exponent depends on the environment in which a sensor network is deployed,
existing techniques on path loss exponent estimation rely on both RSS measurements and distance measurements (obtained
using a method other than from the RSS, e.g., by using a ruler) in the same environment to calibrate the path loss exponent
[1], [13]. RSS measurements are readily available. However distance measurements can be difficult and expensive to obtain in
some environments. Moreover, the reliance on distance measurements impedes deployment in unknown environments.
In this paper we propose several techniques for online calibration of the path loss exponent, which do not rely on distance
2
measurements. The proposed technique also does not assume knowledge of the quantity σdB
in Eq. 1. We demonstrate that
it is possible to estimate the PLE using only power measurements and the geometric constraints associated with planarity
in a sensor network. This may have a significant impact on distance-based wireless sensor network localization. For ease of
explanation, this paper focuses on path loss exponent estimation in 2D. However the proposed technique may also be extended
to 3D.
The rest of the paper is organized as follows. In Section II, we present a technique which estimates the path loss exponent
based on known probability distribution of inter-sensor distances. In Section III we present the principle of the path loss
exponent estimation using the Cayley-Menger determinant [15], [16]. Section IV and Section V present the proposed algorithm
and an improvement of it using data fusion. Both algorithms do not rely on knowledge of distance-related information. The
validity of these algorithms is established using simulation and real measurement data. Finally, conclusions and suggestions
for further work are summarized in Section VI.
II. PATH L OSS E XPONENT E STIMATION BASED ON K NOWN P ROBABILITY D ISTRIBUTION OF I NTER - SENSOR D ISTANCES
In this section, we consider the estimation of the path loss exponent assuming that the probability distribution of distance
between neighboring sensors is known. In some applications, it is possible to know a priori the probability distribution
of distance between neighboring sensors. For example, Stoleru et al. [17] consider that in some applications like precision
agriculture, sensors will be deployed approximately on the grid points with some error. In that case, the approximate probability
distribution of distance between neighboring sensors can be obtained by assuming a certain distribution of the error, e.g.,
Gaussian distribution. Some other researchers also consider the deployment of sensors near grid points [18], [19]. In addition,
it is possible to obtain an accurate knowledge of the probability distribution of inter-sensor distances from the distribution of
wireless sensors [20]–[22]. In particular, it was shown in [20] that under certain conditions, the inter-sensor distance distributions
are very similar when sensors are distributed in the region following a uniform distribution or a Gaussian distribution.
If the distribution has a simple form allowing straightforward computation, the method of moments estimator [23, Section
12.6] can be used to obtain the path loss exponent estimate. For example, if the distance between neighboring nodes is uniformly
distributed in the range [a, b], using Eq. 1 and Eq. 2 the expected value of the measured power can be obtained:
Z b
E(Pij [dBm]) =
P0 [dBm] − 10 × α × log10 xdx
(3)
a
10 × α
b
[−x + xlnx]a .
(4)
ln10
An estimate of α can be obtained by equating the expected value of Pij [dBm] with the sample mean of the measured received
power. In this paper, we assume that all sensors radiate at the same power level for simplicity, i.e., P0 [dBm] is the same for
all sensors. However the derivation in this paper does not rely on this assumption but rather the weaker assumption that the
receiver knows the exact value of the radiated power of the transmitter.
A more generic technique, based on the same idea as the quantile-quantile (q-q) plot [24], can be used if the distance
distribution has a complex form which is not suitable for computation. The q-q plot is a graphical technique for determining
whether two data sets come from populations with a common distribution. Using an approach similar to the q-q plot, a set
of points λ1 , λ2 , ..., λN uniformly distributed in the interval (0, 1) is selected. The quantile points of the distance distribution
=
P0 [dBm](b − a) −
3
d1 , d2 , ..., dN and the measured received power P1 [dBm], P2 [dBm], ..., PN [dBm] corresponding to λ1 , λ2 , ..., λN can then be
determined, where
di = arg {P r(d ≥ di ) = λi },
(5)
and
Pi [dBm] = arg {P r(P [dBm] ≤ Pi [dBm]) = λi }.
(6)
An estimate of α can be obtained by minimizing the function f (α̂):
f (α̂) =
N
X
(Pi [dBm] − P0 [dBm] + 10 × α̂log10 di )2 .
(7)
i=1
Differentiating f (α̂) and setting it to be zero, the value of α̂ can be obtained:
PN
− i=1 (Pi [dBm] − P0 [dBm])log10 di
α̂ =
.
PN
2
i=1 10(log10 di )
(8)
Fig. 1 shows the variation of the mean square error of the estimated path loss exponent α̂ with the number of power
measurements. The mean square error is obtained by running the simulation 100 times with the same number of power
measurements but using different random seed. The parameters used in the simulation are: the true value of α is 2.3; σdB =3.92;
P0 [dBm] = −37.4603dBm; d0 = 1m. The distances dij are uniformly distributed in the interval [1, 18]. These parameters are
drawn from real measurement data reported in [1]. As shown in the figure, this method is very accurate and α̂ converges to
the true value very quickly. When the number of power measurements is greater than 200, the mean square error drops below
0.003.
−1
Mean square error of the estimated path loss exponent
10
−2
10
−3
10
−4
10
Fig. 1.
0
200
400
600
800
1000
1200
1400
Number of power measurements
1600
1800
2000
Variation of the mean square error of α̂ with the number of power measurements
The result shown in Fig. 1 is obtained based on the assumption that the probability distribution of distance between
neighboring sensors is known accurately. This may not be the case in real applications. Further work needs to be done on
investigating the sensitivity of the aforementioned technique with respect to inaccurate knowledge of the probability distribution
of inter-sensor distances.
III. G ENERAL P RINCIPLE OF PATH L OSS E XPONENT E STIMATION USING THE C AYLEY-M ENGER D ETERMINANT
In the last section, we presented a path loss estimation technique based on the knowledge of the probability distribution
of inter-sensor distance. It can calibrate the path loss exponent accurately. However, the proposed technique relies on the
knowledge of distance distribution, which can be unrealistic to obtain in some applications. This disadvantage of the proposed
technique motivates us to find a more generic technique for path loss exponent estimation without relying on knowledge of
the distance distribution, let alone the distance itself. The techniques we present in the following sections also do not assume
2
knowledge of the quantity σdB
in Eq. 1.
4
Consider a sensor with three neighbors and the neighbors of that sensor are also neighbors of each other. Two sensors i and j
are neighbors if Pij is nonzero. The sensor and its neighbors can be represented by a full-connected planar quadrilateral shown
in Fig. 2. The tuple (Pij , dij ) represents the measured power and the true distance between node i and node j respectively. Pij
and dij are related through Eq. 1 and Eq. 2. The path loss exponent estimation problem can be formulated as the simultaneous
p2
P23 , d 23
P12 , d12
P24 , d 24
P13 , d13
p1
p3
P14 , d14
P34 , d 34
p4
Fig. 2.
A fully-connected planar quadrilateral in sensor networks.
estimation of the true distances d12 , d13 , d14 , d23 , d24 , d34 and α given the power measurements P12 , P13 , P14 , P23 , P24 , P34 .
Using the maximum likelihood estimator (MLE), the likelihood function can be obtained as:
Y
(Pij [dBm] − P0 [dBm] + 10αlog10 dij )2
1
L(d12 , d13 , d14 , d23 , d24 , d34 , α) = √
exp(−
).
(9)
2
σdB
( 2πσdB )6
1≤i<j≤4
Maximizing L is equivalent to minimizing g(d12 , d13 , d14 , d23 , d24 , d34 , α), where
X
g(d12 , d13 , d14 , d23 , d24 , d34 , α) =
(Pij [dBm] − P0 [dBm] + 10αlog10 dij )2 .
(10)
1≤i<j≤4
As the number of power measurements is smaller than the number of parameters to be estimated, minimizing g gives us a set
of simple equations. In these equations Pij and P0 are in decimal units, not in dB units, so that Pij = 10Pij [dBm]/10 .
Pij = P0 × d−α
ij , 1 ≤ i < j ≤ 4.
(11)
Another constraint that is required to solve the above equations can be found from the geometric constraint on a fully-connected
planar quadrilateral using the Cayley-Menger determinant [15], [16]. The Cayley-Menger determinant of a quadrilateral is given
by:
¯
¯
¯ 0 d212 d213 d214 1 ¯
¯
¯ 2
¯ d12 0 d223 d224 1 ¯
¯
¯ 2
(12)
D(p1 , p2 , p3 , p4 ) = ¯¯ d13 d223 0 d234 1 ¯¯
¯ d214 d224 d234 0 1 ¯
¯
¯
¯ 1
1
1
1 0 ¯
A classical result on the Cayley-Menger determinant is given by the following theorem:
Theorem 1: (Theorem 112.1 in [16]) Consider an n-tuple of points p1 , ..., pn in m-dimensional space with n ≥ m + 1. The
rank of the Cayley-Menger matrix M (p1 , ..., pn ) (defined analogously to the right side of Eq. 12 but without the determinant
operation) is at most m + 1.
A direct application of the theorem leads to that, in 2D
D(p1 , p2 , p3 , p4 ) = 0.
Combining Eq. 13 and Eq. 11, a nonlinear equation
¯
¯
0
¯
¯ −2/α
C
¯ 12
¯ −2/α
h(α) = ¯ C13
¯ −2/α
¯ C
¯ 14
¯
1
(13)
for α can be obtained:
−2/α
C12
0
−2/α
C23
−2/α
C24
1
−2/α
C13
−2/α
C23
0
−2/α
C34
1
−2/α
C14
−2/α
C24
−2/α
C34
0
1
1
1
1
1
0
¯
¯
¯
¯
¯
¯
¯ = 0,
¯
¯
¯
¯
(14)
5
where Cij = Pij /P0 , 1 ≤ i < j ≤ 4, and the Cij are all known. An analytical solution to Eq. 14 is difficult to find. However
Eq. 14 can be conveniently solved using the bracketing and bisection numerical technique [25, Section 9.1]. The correctness
of the numerical technique has been validated by removing the noise term from Eq. 1, the numerical technique can give the
exact value of α. A gradual increase in the standard deviation of the noise term results in a larger and larger estimation error.
Unfortunately, the estimate of α obtained using this method shows a strong bias, i.e., the bias
Bα̂ = E(α̂) − α
(15)
is certainly nonzero and not a small fraction of α. Fig. 3 shows a histogram of the estimated path loss exponent. The same
parameters used in the last section have been used here. The figure is obtained from 10, 000 different quadrilaterals whose
vertices are uniformly distributed in a square region of 15 × 15. It should be noted that in the presence of noise in power
measurements, Eq. 14 may have non-unique solutions even in the typical range of α for some quadrilaterals. In that case, we
simply discard that quadrilateral and do not use it in the calculation to avoid any ambiguity. In the various simulations shown
in this paper, the number of quadrilaterals discarded accounts for about 15% of the total number of quadrilaterals. For the
example shown in the figure, the true value of α is 2.3 and the estimated value of α has a bias of 2.08. The simulation shows
that α̂ obtained using Eq. 14 has a strong bias which cannot be ignored.
Number of estimated path loss exponent falling into the region
3500
3000
2500
2000
1500
1000
500
0
0
2
4
6
8
Estimated path loss exponent
10
12
14
Fig. 3. Histogram of the estimated path loss exponent using Cayley-Menger determinant. The figure is obtained using 10, 000 quadrilaterals whose vertices
are uniformly distributed in a square region of 15 × 15.
Bias removal requires an analysis on E(α̂). However a direct analysis of E(α̂) is difficult because of the difficulty in
obtaining an explicit analytical expression of α from Eq. 14. Therefore we resort to numerical experiments to evaluate E(α̂).
Fig. 4 shows the relationship between the bias of α̂ (i.e., Bα̂ ), the standard deviation of noise in power measurement σdB , and
α. Fig. 5 shows the corresponding relationship between E(α̂), σdB , and α. Both figures are obtained using 3, 000 quadrilaterals
whose vertices are uniformly distributed in a square of 15 × 15. The power measurements are obtained using Eq. 1 and Eq.
2. Fig. 6 shows the relationship between σα̂ , σdB , and α. The parameter σα̂ is the square root of the sample variance of α̂.
With a little bit abuse of terminology, we call σα̂ the standard deviation of α̂.
Fig. 7 shows the relationship between E(α̂) and σdB at specific values of α = 2.0, α = 3.0 and α = 4.0. Fig. 8 shows the
relationship between σα̂ and σdB at specific values of α = 2.0, α = 3.0 and α = 4.0.
Further simulations show that the relationships between E(α̂), σdB and α, and between σα̂ , σdB and α, are almost
independent of the distribution of the vertices of the quadrilaterals, and independent of the shape of the area in which the
vertices of the quadrilaterals are located. These are shown in Fig. 9 and Fig. 10. Fig. 9 shows the relationship between E(α̂),
σdB and α for quadrilaterals whose vertices are located in areas of a variety of different shapes and distributed in the area
following different distributions. This relationship between E(α̂), σdB and α is explored in Section IV to approximately correct
the bias in E(α̂). Fig. 10 shows the relationship between σα̂ , σdB and α. This relationship between σα̂ , σdB and α is further
explored in Section V.
IV. PATH L OSS E XPONENT E STIMATION BASED ON PATTERN M ATCHING
Based on the observation shown in Fig. 9 that the relationship between E(α̂), σdB and α is independent of the distribution
of the vertices of various quadrilaterals and the shape of the area in which vertices of the quadrilaterals are located, a pattern
matching technique can be used to estimate the path loss exponent using the power measurements only.
6
20
Bias of the PLE Estimation
18
16
14
12
10
8
6
4
2
30
25
4
20
3.5
15
3
10
2.5
5
2
Standard Deviation of Noise in Power (dB)
Relationship between the bias of α̂, the standard deviation of noise in power measurement σdB and α. Bα̂ depends on α as well as on σdB .
Mean Value of the Estimated Path Loss Exponent
Fig. 4.
True Path Loss Exponent
22
20
18
16
14
12
10
8
6
4
30
25
20
15
10
5
2
Standard Deviation of Noise in Power (dB)
Fig. 5.
2.2
2.4
2.6
2.8
3
3.2
3.4
3.6
3.8
4
True Path Loss Exponent
Relationship between E(α̂), the standard deviation of noise in power measurement σdB and α. E(α̂) depends on α as well as on σdB .
´ simulation, a data base can be established where each entry in the data base is arranged in the form
³ Specifically, via a priori
2
2
2
. The symbol E(α̂)αi ,σdB,j
is used to emphasize dependence of E(α̂) on the path loss exponent αi
αi , σdB,j
, E(α̂)αi ,σdB,j
2
and the variance of noise in power measurements σdB,j . This data base can be obtained using a large number of quadrilaterals
whose vertices, say, are uniformly distributed in an area. The corresponding power measurements are obtained using Eq. 1
2
and Eq. 2. The distance between adjacent σdB
is the same, i.e.,
2
2
2
σdB,j+1
− σdB,j
= ∆σdB
.
(16)
The distance between adjacent α is also the same, i.e.,
αi+1 − αi = ∆α.
(17)
2
and ∆α are empirically chosen constants, which do not change with i and j.
The parameters ∆σdB
Given this data base, the estimation of α can be obtained using the following procedure for pattern matching:
1) Identify a set of fully connected quadrilaterals in the wireless sensor network for further computation;
2
2) Add a random Gaussian noise with variance ∆σdB
into each power measurement. The power is measured in dB milliwatts
unit;
Standard deviation of the Path Loss Exponent Estimation
7
9
8
7
6
5
4
3
2
1
30
25
4
20
3.5
15
3
10
2.5
5
Standard Deviation of Noise in Power (dB)
2
True Path Loss Exponent
Fig. 6. Relationship between the standard deviation of α̂, the standard deviation of noise in power measurement σdB and α. σα̂ depends on σdB , and to a
small degree on α as well. At larger values of σdB , σα̂ is almost independent of α. The dependence of σα̂ on α is more significant with smaller values of
σdB .
25
α =2
α =3
α =4
Mean Value of the Estimated Path Loss Exponent
20
15
10
5
0
Fig. 7.
0
5
10
15
20
Standard Deviation of Noise in Power (dB)
25
30
Relationship between E(α̂) and the standard deviation of noise in power measurement σdB at specific values of α = 2.0, α = 3.0 and α = 4.0.
2
can be obtained as the
3) For each individual quadrilateral, solve for α̂ using Eq. 14. An E(α̂)r,1 corresponding to ∆σdB
average value of α̂ obtained from each individual quadrilateral. Here the subscript r is used to mark the difference with
the corresponding value in the database;
2
4) Repeat steps 2 and 3 using different noise variance values M ∆σdB
, where M = 0, 1, 2, ..., m. Here M = 0 corresponds
2
to the original data set without the additionally introduced noise. A series of tuples (0, E(α̂)r,0 ), (∆σdB
, E(α̂)r,1 ), ...,
2
(m∆σdB , E(α̂)r,m ) can then be obtained;
5) Search the database and find the values of i and j such that:
{i, j} = arg min{i,j}
m
X
2
(E(α̂)r,N − E(α̂)αi ,σdB,j+N
)2 .
(18)
N =0
The parameter m has to be a large number, say ≥ 500, in order to obtain a reasonably accurate estimate of α, which is
robust against the randomness in the data. As will be shown in the next section, the accuracy of the path loss exponent
2
estimation can be increased by choosing a larger value of m and smaller values of ∆σdB
and ∆α.
8
10
α =2
α =3
α =4
9
8
7
6
5
4
3
2
1
0
0
5
10
15
20
Standard Deviation of Noise in Power (dB)
25
30
Fig. 8. Relationship between the standard deviation of α̂ and the standard deviation of noise in power measurement σdB at specific values of α = 2.0,
α = 3.0 and α = 4.0.
6) Finally, an improved estimate of α approximately correcting for the bias can be obtained as:
α̂ = αi .
(19)
σ̂dB = σdB,j .
(20)
Similarly, an estimate of σdB can be obtained as:
A. Simulation Validation
In this section, we shall validate the proposed technique using both simulations and real measurements. The database is
established by simulations using 3, 000 quadrilaterals whose vertices are uniformly distributed in a square region of 15 × 15.
2
The measured power is generated using Eq. 1 and Eq. 2. ∆σdB
is set to be 1 and the distance between adjacent αi is set to
be 0.1, i.e., ∆α = 0.1. Another set of 3, 000 quadrilaterals whose vertices is uniformly distributed in a rectangular area of
10 × 20 are used to establish the performance of the proposed algorithm. 700 points are used in the search, i.e., m = 700 in
Eq. 18. The simulation is repeated by varying the value of α from 2 to 4 and varying the value of σdB from 5 to 10, which
are the typical ranges of the two parameters [11]. Fig. 11 shows the empirical cumulative distribution function (CDF) of error
in estimating α and Fig. 12 shows the the empirical CDF of error in estimating σ.
As shown in Fig. 11, the error in estimating α is contained in the region [−0.3, 0.1]. The mean estimation error is −0.1175
and the error variance is 0.0115. Fig. 12 shows that the error in estimating σdB is contained in the region [0, 1]. The mean
estimation error is 0.4827 and the error variance is 0.0263. The estimation error is attributable to the randomness in the data
2
but can be reduced by using smaller values of ∆σdB
and ∆α at the expense of increased computational load. For example,
2
our simulation shows that by using m = 2800, ∆α = 0.05 and ∆σdB
= 0.25, the maximum error in estimating α reduces to
0.2. The mean error reduces to −0.0603 and the error variance reduces to 0.0056.
Simulations using quadrilaterals whose vertices are distributed in an area of a different shape and/or following a different
distribution show similar performance. Fig. 13 and Fig. 14 show the simulation results using a set of 3, 000 quadrilaterals
whose vertices are distributed in a square region of 15 × 15 following a truncated two-dimensional Gaussian distribution. The
mean of the Gaussian distribution is at the center of the square region and the standard deviation of the Gaussian distribution
is 5. Both figures show similar performance as those in Fig. 11 and Fig. 12 except that the estimation error of α has a value
of −0.4 at a couple of points in Fig. 13. The mean error in estimating α is −0.1785. The estimation error of σdB is better
than that in Fig. 12 and the mean estimation error is −0.1252. Fig. 15 and 16 show the simulation results using a set of 3, 000
quadrilaterals whose vertices are distributed in a rectangular region of 10 × 15 following a Poisson distribution. The mean
error in estimating α is −0.0113 and the mean estimation error of σdB is 0.1021.
B. A Comparison with a MLE of the PLE Using both Power and Distance Measurements
To further evaluate the performance of the proposed technique, we apply the proposed PLE estimation technique to real
measurement data and compare the PLE estimate obtained using the proposed technique with that in [1] which obtains a
maximum likelihood estimate of the PLE using both power measurements and distance measurements. The measurement data
22
Mean Value of the Estimated Path Loss Exponent
Mean Value of the Estimated Path Loss Exponent
9
20
18
16
14
12
10
8
6
4
30
22
20
18
16
14
12
10
8
6
4
30
25
20
15
10
Standard Deviation of Noise in Power (dB)
5
2
2.2
2.4
2.6
2.8
3
3.2
3.4
3.6
3.8
25
4
20
15
10
Standard Deviation of Noise in Power (dB) 5
True Path Loss Exponent
2
2.4
2.6
2.8
3.2
3.4
3.6
4
3.8
True Path Loss Exponent
(b)
(a)
22
Mean Value of the Estimated Path Loss Exponent
Mean Value of the Estimated Path Loss Exponent
2.2
3
20
18
16
14
12
10
8
6
4
30
22
20
18
16
14
12
10
8
6
4
30
25
20
15
10
Standard Deviation of Noise in Power (dB)
5
2
(c)
2.2
2.4
2.6
2.8
3
3.2
3.4
True Path Loss Exponent
3.6
3.8
4
25
20
15
10
Standard Deviation of Noise in Power (dB)
5
2
2.2
2.4
2.6
2.8
3
3.2
3.4
3.6
3.8
4
True Path Loss Exponent
(d)
Fig. 9. Relationship between the bias of E(α̂), σdB and α. The relationship between E(α̂), σdB and α is almost independent of the distribution of
the vertices of the quadrilaterals and independent of the shape of the area in which the vertices of the quadrilaterals are located. a) Generated from 3000
quadrilaterals whose vertices are uniformly distributed in a “L” shaped area. b) Generated from 3000 quadrilaterals whose vertices are uniformly distributed
in a rectangular area of 10x20. c) Generated from 3000 quadrilaterals whose vertices are distributed in a square area of 15x15 following a truncated Gaussian
distribution with a zero mean and a standard variation of 5. d) Generated from 3000 quadrilaterals whose vertices are located in a ring. The inner radius of
the ring is 5 and the outer radius is 10. The coordinates of the vertices are generated by first selecting a number uniformly distributed in [5, 10] and then
selecting a number uniformly distributed in [0, 2π]. The two numbers are used as the polar coordinate of the vertex to obtain the rectangular coordinate.
and the deployment of sensors can also be found at http://www.eecs.umich.edu/∼hero/localize/. The wireless sensor network
in [1] consists of 44 fully connected nodes, which make up 135, 751 quadrilaterals. It is both computationally intensive and
unnecessary to compute α̂ for each quadrilateral. Here we randomly choose 10, 000 quadrilaterals for computation. The reference
power P0 (d0 )[dBm] is calculated using the free space Friis equation at a reference distance d0 = 1m and P0 (d0 )[dBm] =
−37.4663dBm [1]. The PLE estimated using the proposed technique is 2.2. Using both power measurements and distance
measurements (946 distance measurements and 946 power measurements from 44 full connected nodes), a maximum likelihood
estimate of the PLE is shown to be 2.3022 [1]. Assuming the PLE estimate obtained using both power measurements and
distance measurements represents the true value of PLE, the PLE estimate obtained using the proposed technique which does
not use distance measurements has an estimation error of −0.1022. The true value of σdB (i.e., the value obtained using both
power measurements and measured distances) is 3.92 and the estimation error is −1.50. The slightly larger error in estimating
σdB when using real measurement data may be attributable to a deviation of the density of the noise in the power measurements
from a Gaussian distribution.
As a further comparison of the proposed technique with the maximum likelihood estimator using both power and distance
measurements, we analyze the number of distance measurements required by the maximum likelihood estimator to make a
PLE estimate with a similar accuracy as the proposed technique. Given k distance measurements d1 , d2 , · · ·, dk (recall that all
distances are normalized with regards to known distance d0 ) and the corresponding power measurements P1 [dBm], P2 [dBm],
· · ·, Pk [dBm], it can be readily shown from Eq. 1 and Eq. 2 that a maximum likelihood estimate of the PLE is:
Pk
− i=1 (Pi [dBm] − P0 [dBm])log10 di
.
(21)
α̂m =
Pk
10 i=1 (log10 di )2
Standard deviation of the Path Loss Exponent Estimation
Standard deviation of the Path Loss Exponent Estimation
10
9
8
7
6
5
4
3
2
1
30
9
8
7
6
5
4
3
2
1
30
25
20
15
3
10
Standard Deviation of Noise in Power (dB)
5
2
2.2
2.4
2.6
3.2
3.4
3.6
3.8
25
4
20
15
10
Standard Deviation of Noise in Power (dB)
2.8
True Path Loss Exponent
5
2
2.4
2.6
2.8
3.4
3.2
3.6
4
3.8
True Path Loss Exponent
(b)
Standard deviation of the Path Loss Exponent Estimation
(a)
Standard deviation of the Path Loss Exponent Estimation
2.2
3
9
8
7
6
5
4
3
2
1
30
9
8
7
6
5
4
3
2
1
30
25
20
15
10
Standard Deviation of Noise in Power (dB)
5
2
2.2
2.4
2.6
2.8
3
3.2
3.4
3.6
3.8
25
4
20
15
10
Standard Deviation of Noise in Power (dB)
True Path Loss Exponent
5
2
2.2
2.4
2.6
2.8
3
3.2
3.4
3.6
3.8
4
True Path Loss Exponent
(d)
(c)
Fig. 10. Relationship between σα̂ , σdB and α. The relationship between between σα̂ , σdB and α is almost independent of the distribution of the vertices
of the quadrilaterals and independent of the shape of the area in which the vertices of the quadrilaterals are located. Sub-figures a), b), c), d) are obtained
under the same conditions as those in Fig. 9.
According to Eq. 1 and Eq. 2, Pi [dBm] is related to di by:
Pi [dBm] = P0 [dBm] − 10αlog10 di + ni ,
(22)
where the sequence {ni } is a stationary zero mean white Gaussian sequence of random variables with standard deviation σdB .
Replacing Pi [dBm] in Eq. 21 with Eq. 22, it can be shown that
Pk
i=1 ni log10 di
α̂m = α −
,
(23)
Pk
10 i=1 (log10 di )2
and
E(α̂m − α)2 =
100
Pk
2
σdB
i=1 (log10 di )
2
.
(24)
Eq. 24 shows that the mean square error of the estimated PLE depends on the distribution of distances. Considering a simple
scenario of k distance measurements uniformly distributed between 1 and 10, i.e., di = 1 + 9i/k, Eq. 24 becomes
E(α̂m − α)2 =
100
Pk
2
σdB
i=1 (log10 (1
+ 9i/k))2
.
(25)
Fig. 17 shows the variation of the mean square error of the estimated PLE with σdB and number of distance measurements.
Using the performance of the proposed technique shown in Fig. 11 for example, the mean square error of the estimated PLE,
i.e., the sum of the square of the mean estimation error and the error variance, is 0.025. When σdB is equal to 5, the maximum
likelihood estimator using both power and distance measurements needs 18 distance measurements to achieve a similar accuracy
as the proposed technique. A larger number of distance measurements is required at a larger value of σdB and the converse.
In addition to not requiring distance measurements, another advantage of the proposed technique is it makes the PLE
estimation possible in hostile, dangerous or inaccessible environments. Note that even in the same environment, the channel
11
Empirical CDF
1
0.9
CDF of the Path Loss Exponent Estimation Error
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
−0.35
−0.3
−0.25
−0.2
−0.15
−0.1
−0.05
Path Loss Exponent Estimation Error
0
0.05
0.1
0.15
Fig. 11. The empirical CDF of error in estimating α using the pattern matching technique. The vertices of the quadrilaterals are uniformly distributed in a
rectangular area of 10 × 20.
Empirical CDF
1
0.9
0.8
CDF of the σ
dB
Estimation Error
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
σ Estimation Error
0.7
0.8
0.9
1
dB
Fig. 12. The empirical CDF of error in estimating σdB using the pattern matching technique. The vertices of the quadrilaterals are uniformly distributed in
a rectangular area of 10 × 20.
characteristics may change considerably over a long period of time due to seasonal changes and weather changes [11], the
proposed technique also show advantage for implementation in these environments.
C. A Discussion on the Computational Complexity and the Impact of the PLE Estimation Error
In this section, we have a brief discussion of the computational complexity of the proposed technique and the impact of the
PLE estimation error on sensor network localization.
To evaluate the computational complexity of the proposed algorithm, further simulations are performed to estimate the
number of quadrilaterals required in order to obtain a stable performance. The simulation results are shown in Fig. 18. Fig. 18
shows that a minimum of 1000 quadrilaterals are required in order for the proposed algorithm to achieve a stable performance.
In the proposed algorithm, the data base required for pattern matching can be established via a priori simulation. The
memory space required for storing the data base is in the order of hundreds of kilo-bytes which is not a major constraint for
the implementation of the proposed algorithm. However the online computation involving 1000 quadrilaterals and the pattern
matching may present a major challenge for distributed implementation of the proposed algorithm. Therefore the proposed
algorithm in its present form is not suitable for implementation in a distributed environment. In a centralized environment,
the proposed algorithm can be implemented in a central station or in a cluster head which has more energy and computation
power than an ordinary sensor node. It remains a future research topic to design a distributed algorithm in which each sensor
12
Empirical CDF
1
0.9
CDF of the Path Loss Exponent Estimation Error
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
−0.5
−0.4
−0.3
−0.2
−0.1
Path Loss Exponent Estimation Error
0
0.1
0.2
Fig. 13. The empirical CDF of error in estimating α using the pattern matching technique. The vertices of the quadrilaterals are distributed in a square
region of 15 × 15 following a truncated two-dimensional Gaussian distribution.
Empirical CDF
1
0.9
0.8
CDF of the σ
dB
Estimation Error
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
−0.6
−0.5
−0.4
−0.3
σ
dB
−0.2
Estimation Error
−0.1
0
0.1
0.2
Fig. 14. The empirical CDF of error in estimating σdB using the pattern matching technique. The vertices of the quadrilaterals are distributed in a square
region of 15 × 15 following a truncated two-dimensional Gaussian distribution.
does the computation for the quadrilateral it resides in and an improved estimate of PLE is obtained by fusing the computation
results of individual sensors.
The impact of the PLE estimation error on the estimation of individual inter-sensor distances can be readily evaluated. The
answer to the more involved question of the impact of the PLE estimation error on sensor network localization is actually
linked to a fundamental question: what is the impact of error in distance measurements on the performance of distance-based
localization algorithm? Numerous localization algorithms have shown via simulations that the impact of distance measurement
error possibly depends on a number of factors which include the distribution of anchors (i.e, sensor nodes with known positions)
and non-anchor (i.e., sensor nodes whose positions are to be estimated) nodes, node degree (or equivalently node density and
sensing range) and the shape of the area in which sensors are deployed [14]. We are yet to develop an accurate knowledge in
the area. In this paper instead of giving a comprehensive evaluation of the impact of PLE estimation error on sensor network
localization, we give an illustration of the possible impact of the PLE estimation error using the measurement data in [1]. The
wireless sensor network in [1] consists of 44 fully connected nodes. The four sensors located at the four corners are chosen
to be the anchor nodes and the rest are the non-anchor nodes. Fig. 19 shows the placement of these sensors. The true value
of PLE is 2.3022 [1]. An error term varying from −0.4 to 0.4 is added to the true value to evaluate the impact of the PLE
estimation error. The sum of the true PLE and the error is termed as the estimated PLE, i.e., α̂. The sensor network localization
13
Empirical CDF
1
0.9
CDF of the Path Loss Exponent Estimation Error
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
−0.25
−0.2
−0.15
−0.1
−0.05
0
0.05
Path Loss Exponent Estimation Error
0.1
0.15
0.2
0.25
Fig. 15. The empirical CDF of error in estimating α using the pattern matching technique. The vertices of the quadrilaterals are drawn randomly from nodes
distributed in a rectangular region of 10 × 15 following a Poisson distribution with a uniform node density of 100/(10 × 15).
Empirical CDF
1
0.9
0.8
CDF of the σ
dB
Estimation Error
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
−0.3
−0.2
−0.1
0
0.1
0.2
σ Estimation Error
0.3
0.4
0.5
0.6
dB
Fig. 16. The empirical CDF of error in estimating σdB using the pattern matching technique. The vertices of the quadrilaterals are drawn randomly from
nodes distributed in a rectangular region of 10 × 15 following a Poisson distribution with a uniform node density of 100/(10 × 15).
is formulated as the following optimization problem:
{x̂i, m<i≤n } = arg min{x̂i,m<i≤n }
n
X
X
(Pij − 10α̂log10 ||x̂i − x̂j ||)2 ,
(26)
i=m+1 j∈Ni
where anchor nodes are numbered from 1 to m, non-anchor nodes are numbered from m+1 to n, Ni denotes the set of
neighbors of node i and || · || denotes the Euclidean norm. x̂i and xi are the estimated position and the true position of the ith
sensor respectively and for anchor nodes x̂i = xi . The simulated annealing algorithm [26] is used to solve the problem and
obtain the estimated positions of non-anchor nodes. Fig. 20 shows the variation of the sensor network position estimation error
with the PLE estimation error. As shown in the figure, the sensor network localization appears to be more robust against an
overestimation of the PLE than an underestimation. Actually, with an overestimation of the PLE, we find a slight improvement
in the sensor network localization performance. A plausible explanation of the simulation result is:
• First, the MLE of inter-sensor distance in Eq. 11 is a biased estimate [14] and it tends to overestimate the distance even
when the PLE is accurate. A slightly overestimate of the PLE may help to balance this effect.
• Second, this may also be attributable to the placement of anchors and the high node degree of anchors. Given a set
of power measurements, an overestimate of the PLE will result in smaller inter-sensor distance estimates. Therefore a
“shrinkage” of the estimated sensor positions may occur. However because the anchors are placed at the four corners of
14
0
10
σ =5
dB
σ =6
dB
σ =7
dB
σdB=8
σ =9
dB
σ =10
Mean square error of α estimate
dB
−1
10
−2
10
−3
10
0
10
20
30
40
50
60
Number of distance measurements
70
80
90
100
Fig. 17. Variation of the mean square error of the estimated PLE with σdB and number of distance measurements. The PLE is obtained from a maximum
likelihood estimator using both power and distance measurements.
0.8
Maximum estimation error
Mean estimation error
Variance of estimation error
0.7
0.6
Error metric
0.5
0.4
0.3
0.2
0.1
0
−0.1
0
500
1000
1500
Number of nodes
2000
2500
3000
Fig. 18. Variation of the maximum estimation error of α̂, mean estimation error and error variance with number of quadrilaterals used in the simulation.
Each point shown in the figure is the average value of ten simulations using different random seed. The vertices of the quadrilaterals are drawn randomly
from nodes distributed in a rectangular region of 10 × 15 following a Poisson distribution with a uniform node density of 100/(10 × 15).
the map and they have a high node degree, they constrain the shrinking effect caused by an overestimation of the PLE.
This constraint formed by placing the anchors at the corners becomes less effective with an underestimation of the PLE.
V. F URTHER I MPROVEMENT ON THE P ROPOSED T ECHNIQUE USING DATA F USION
In Section IV, we presented a technique which approximately corrects the bias of α̂ using the relationship between E(α̂),
σdB and α. Unsurprisingly a similar technique can also be developed, which begins by establishing a database similar to that
in Section IV but describing the relations between σα̂ (instead of E(α̂)), σdB and α, and uses the pattern matching technique
to correct the bias of α̂ based on the relationship between σα̂ , σdB and α. Fig. 21 shows the error in estimating α. The mean
estimation error is −0.2597 and the error variance is 0.1295. The same parameters as used in Section IV are used here.
As shown in the figure, the error in estimating α is significantly worse than that shown in Section IV. This result is expected
because in comparison with E(α̂), the value of σα̂ is less dependent on the value of α, as suggested in Fig. 7 and 8. However
the relationship between σα̂ , σdB and α does provide extra information concerning α, which can be exploited to improve α̂
obtained in Section IV by using data fusion [27].
A popular approach in data fusion is to linearly combine the estimated parameters. Denote the estimated α using the
relationship between σα̂ , σdB and α by α̂s . Denote the estimated α using the relationship between E(α̂), σdB and α by α̂b .
15
14
Non−anchors
Anchors
12
10
8
6
4
2
0
−2
−5
Fig. 19.
0
5
10
True positions of anchors and non-anchor nodes in the simulation.
25
Sensor network position estimation error
20
15
10
5
0
−0.4
−0.3
−0.2
−0.1
0
0.1
Path loss exponent estimation error
0.2
0.3
0.4
Fig. 20. Variation
position estimation error with the PLE estimation error. The sensor network position estimation error shown in the
Pn of the sensor network
1
2
th sensor respectively, sensors 1 to m are
figure is n−m
i=m+1 ||x̂i − xi || , where x̂i and xi are the estimated position and the true position of the i
anchors, and sensors m+1 to n are non-anchors. Each point shown in the figure is the average value of 10 simulations using different random seed.
The dual estimated vector is defined as:
The estimated fusion calculation is given by:
→
−
α = [α̂s α̂b ]T .
(27)
→
α̂ = WT −
α,
(28)
where W is a vector in our specific application. In the more general case of simultaneous estimation of multiple parameters,
W is a full rank weight matrix. A typical data fusion approach is to calculate W such that the sum of the diagonal elements
for the error covariance matrix of the fused estimate is minimized. The optimal W, which gives the lowest error covariance
is given by [27]:
¢−1
¡
.
(29)
W = C−1 AT AC−1 AT
where
·
C = E{
α̂s − α
α̂b − α
¸·
α̂s − α
α̂b − α
¸T
},
(30)
and
A = [1 1]
(31)
16
Empirical CDF
1
0.9
CDF of the Path Loss Exponent Estimation Error
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
−1.2
−1
−0.8
−0.6
−0.4
−0.2
Path Loss Exponent Estimation Error
0
0.2
0.4
Fig. 21. The empirical CDF of error in estimating α using the pattern matching technique and the relationship between σα̂ , σdB and α. The vertices of the
quadrilaterals are uniformly distributed in a rectangular area of 10 × 20.
in our specific application. α is the corresponding true value. It should be noted that the optimality of data fusion in Eq. 29
relies on the assumptions that covariance is the dominant factor in estimation error in comparison with the bias, and that the
estimation errors have a Gaussian distribution [27]. These requirements make the optimality of this data fusion method difficult
to prove. However this method is generally robust in the sense that given an accurate error covariance matrix, it always results
in a fused estimator with a lower covariance than the individual estimators [28].
To establish the performance improvement of data fusion approach, we use the path loss exponent estimation from the
quadrilaterals, whose vertices are distributed in a square region of 15 × 15 following a truncated two-dimensional Gaussian
distribution (as shown in Fig. 13), to obtain the error covariance matrix C and the weight vector W. The obtained weight vector
is W = [−0.1718, 1.1718]T . A negative value of −0.1718 indicates that the error in α̂s and the error in α̂b are positively
correlated. Using this weight vector, we linearly combine the path loss exponent estimates obtained using the relationship
between σα̂ , σdB and α, and using the relationship between E(α̂), σdB and α (as shown in Eq. 28).
Empirical CDF
1
0.9
CDF of the Path Loss Exponent Estimation Error
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
−0.4
Fig. 22.
−0.3
−0.2
−0.1
0
Path Loss Exponent Estimation Error
0.1
0.2
0.3
Error in estimating α using data fusion. The vertices of the quadrilaterals are uniformly distributed in a rectangular area of 10 × 20.
Fig. 22 shows the path loss exponent estimation error for quadrilaterals, whose vertices are uniformly distributed in a
rectangular area of 10 × 20. An overall improvement has been achieved in comparison with the results shown in Fig. 11,
although at some points the estimation error becomes larger, which can be possibly explained by the statistical nature of data
fusion. The mean estimation error reduces from the original −0.1175 to −0.0930. The error variance remains essentially the
same.
17
VI. C ONCLUSIONS AND F URTHER W ORK
In this paper, we presented some techniques for online calibration of path loss exponent in wireless sensor networks without
relying on distance measurements. Specifically, techniques were proposed which are based on different assumptions about
knowledge of distance information.
The first technique assumes that the probability distribution of distance between neighboring sensors is known. Then an
algorithm similar to the quantile-quantile plot was proposed, which can estimate the path loss exponent accurately using a small
number of received power measurements. However this assumption of knowing the distance distribution can be unrealistic in
some applications. This has motivated us to find a more generic technique without using any distance information.
Then we presented a technique based on the Cayley-Menger determinant, which estimates the path loss exponent using
only power measurements and the geometric constraints associated with planarity in a wireless sensor network. The technique
can give an accurate estimate of α when there is no noise in power measurements, but it has a large bias in the presence
of noise. A pattern matching technique approximately correcting the bias is proposed based on the empirical observation that
the relationship between E(α̂), σdB and α is independent of the distribution of the vertices of various quadrilaterals and the
shape of the area in which vertices of the quadrilaterals are located. We also presented an improvement of the earlier technique
using data fusion. The proposed algorithms may have significant impact on distance-based wireless sensor network localization,
where distance is estimated from the received signal strength measurements.
In this paper, we observed the empirical law that the relationship between E(α̂), σdB and α is independent of the distribution
of the vertices of the quadrilaterals and is also independent of the shape of the area in which the vertices of the quadrilaterals
are located. It is desirable to obtain an analytical expression of the relationship between E(α̂), σdB and α. This is the direction
of our future research.
Furthermore, the proposed algorithm relies on the log-normal propagation model in Eq. 1 and Eq. 2 in the sense that the
maximum likelihood estimator shown in Eq. 11 may have a different form when the received signal strength has a different
model. Although the log-normal propagation model is a popular model for wireless networks, there are environments in which
the log-normal propagation model is not the best model [11], [12]. In that case, a technique needs to be developed to select the
best model and choose the best estimator for distance to replace Eq. 11 accordingly. Therefore how to develop an algorithm
for environments in which the log-normal propagation model does not apply is also a future research topic.
R EFERENCES
[1] N. Patwari, I. Hero, A.O., M. Perkins, N. Correal, and R. O’Dea, “Relative location estimation in wireless sensor networks,” IEEE Transactions on
Signal Processing, vol. 51, no. 8, pp. 2137–2148, 2003.
[2] P. Bahl and V. Padmanabhan, “Radar: an in-building rf-based user location and tracking system,” in IEEE INFOCOM, vol. 2, 2000, pp. 775–784.
[3] P. Bergamo and G. Mazzini, “Localization in sensor networks with fading and mobility,” in The 13th IEEE International Symposium on Personal, Indoor
and Mobile Radio Communications, vol. 2, 2002, pp. 750–754.
[4] E. Elnahrawy, X. Li, and R. Martin, “The limits of localization using signal strength: a comparative study,” in First Annual IEEE Communications
Society Conference Sensor and Ad Hoc Communications and Networks, 2004, pp. 406–414.
[5] D. Madigan, E. Einahrawy, R. Martin, W.-H. Ju, P. Krishnan, and A. Krishnakumar, “Bayesian indoor positioning systems,” in IEEE INFOCOM 2005,
vol. 2, 2005, pp. 1217–1227.
[6] D. Niculescu and B. Nath, “Localized positioning in ad hoc networks,” in IEEE International Workshop on Sensor Network Protocols and Applications,
2003, pp. 42–50.
[7] P. Prasithsangaree, P. Krishnamurthy, and P. Chrysanthis, “On indoor position location with wireless lans,” in The 13th IEEE International Symposium
on Personal, Indoor and Mobile Radio Communications, vol. 2, 2002, pp. 720–724.
[8] S. Ray, W. Lai, and I. Paschalidis, “Deployment optimization of sensornet-based stochastic location-detection systems,” in IEEE INFOCOM 2005, vol. 4,
2005, pp. 2279–2289.
[9] P. Krishnan, A. Krishnakumar, W.-H. Ju, C. Mallows, and S. Gamt, “A system for lease: location estimation assisted by stationary emitters for indoor
rf wireless networks,” in IEEE INFOCOM, vol. 2, 2004, pp. 1001–1011.
[10] T. Roos, P. Myllymaki, and H. Tirri, “A statistical modeling approach to location estimation,” IEEE Transactions on Mobile Computing, vol. 1, no. 1,
pp. 59–69, 2002.
[11] T. S. Rappaport, Wireless Communications: Principles and Practice, 2nd ed. Prentice Hall PTR, 2001.
[12] K. Pahlavan and A. H. Levesque, Wireless Information Networks, 2nd ed. John Wiley & Sons, 2005.
[13] D. Lymberopoulos, Q. Lindsey, and A. Savvides, “An empirical analysis of radio signal strength variability in ieee 802.15.4 networks using monopole
antennas, Tech. Rep. ENALAB Technical Report 050501, 2005.
[14] G. Mao, B. Fidan, and B. D. O. Anderson, “Localization,” in Sensor Network and Configuration: Fundamentals, Techniques, Platforms and Experiments
(in press). Germany: Springer-Verlag, 2006.
[15] G. M. Crippen and T. F. Havel, Distance Geometry and Molecular Conformation. New York: John Wiley and Sons Inc., 1988.
[16] L. M. Blumenthal, Theory and Applications Distance Geometry. Oxford University Press, 1953.
[17] R. Stoleru and J. Stankovic, “Probability grid: a location estimation scheme for wireless sensor networks,” in First Annual IEEE Communications Society
Conference on Sensor and Ad Hoc Communications and Networks, 2004, pp. 430–438.
[18] S. Dhillon, K. Chakrabarty, and S. Iyengar, “Sensor placement for grid coverage under imprecise detections,” in Information Fusion, 2002. Proceedings
of the Fifth International Conference on, vol. 2, 2002, pp. 1581–1587.
[19] K. Chakrabarty, S. Iyengar, H. Qi, and E. Cho, “Grid coverage for surveillance and target location in distributed sensor networks,” Computers, IEEE
Transactions on, vol. 51, no. 12, pp. 1448–1453, 2002.
[20] L. E. Miller, “Distribution of link distances in a wireless network,” Journal of Research of the National Institute of Standards and Technology, vol. 106,
no. 2, pp. 401–412, 2001.
[21] J. P. Mullen, “Robust approximations to the distribution of link distances in a wireless network occupying a rectangular region,” Mobile COmputing and
Communications Review, vol. 7, no. 2, 2003.
[22] C.-C. Tseng, H.-T. Chen, and K.-C. Chen, “On the distance distributions of the wireless ad hoc networks,” in IEEE VTC Spring, Melbourne, 2006.
18
[23] R. Bartoszynski and M. Niewiadomska-Bugaj, Probability and Statistical Inference, ser. Wiley Series in Probability and Statistics. John Wiley & Sons
Inc., 1996.
[24] R. J. Hyndman and Y. Fan, “Sample quantiles in statistical packages,” Amer. Stat, vol. 50, pp. 361–365, 1996.
[25] W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Receipes in C - The Art of Scientific Computing, 2nd ed. Cambridge
University Press, 2002.
[26] A. A. Kannan, G. Mao, and B. Vucetic, “Simulated annealing based localization in wireless sensor network,” in The 30th IEEE Conference on Local
Computer Networks, 2005, pp. 513–514.
[27] Y. Zhu, Multisensor Decision and Estimation Fusion, ser. The International Series on Asian Studies in Computer and Information Science. Springer,
2002.
[28] M. McGuire, K. N. Plataniotis, and A. N. Venetsanopoulos, “Data fusion of power and time measurements for mobile terminal location,” IEEE Transactions
on Mobile Computing, vol. 4, no. 2, pp. 142–153, 2005, 1536-1233.